Multiple solutions for Schr{ö}dinger-Poisson-Slater equations with critical growth

Abstract

We obtain multiple solutions for the zero mass Schr{ö}dinger-Poisson-Slater equation \[ - Δu + \left( \frac{1}{4 π| x |} \ast u^2 \right) u = λg (x) | u |^{p - 2} u + | u |^{6 - 2} u \text{, \ \ \ \ } u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] for $\hat{I}»\gg 1$, where $p\in (4,6)$ and $g\in L^{6/(6-p)}({\mathbb{R}}^3)$. The crucial (PS)${}_c$ condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.